Computational Mechanics

, Volume 52, Issue 4, pp 959–969 | Cite as

At least three invariants are necessary to model the mechanical response of incompressible, transversely isotropic materials

  • M. Destrade
  • B. Mac Donald
  • J. G. MurphyEmail author
  • G. Saccomandi
Original Paper


The modelling of off-axis simple tension experiments on transversely isotropic nonlinearly elastic materials is considered. A testing protocol is proposed where normal force is applied to one edge of a rectangular specimen with the opposite edge allowed to move laterally but constrained so that no vertical displacement is allowed. Numerical simulations suggest that this deformation is likely to remain substantially homogeneous throughout the specimen for moderate deformations. It is therefore further proposed that such tests can be modelled adequately as a homogenous deformation consisting of a triaxial stretch accompanied by a simple shear. Thus the proposed test should be a viable alternative to the standard biaxial tests currently used as material characterisation tests for transversely isotropic materials in general and, in particular, for soft, biological tissue. A consequence of the analysis is a kinematical universal relation for off-axis testing that results when the strain-energy function is assumed to be a function of only one isotropic and one anisotropic invariant, as is typically the case. The universal relation provides a simple test of this assumption, which is usually made for mathematical convenience. Numerical simulations also suggest that this universal relation is unlikely to agree with experimental data and therefore that at least three invariants are necessary to fully capture the mechanical response of transversely isotropic materials.


Biomechanics Constitutive laws  Finite element 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • M. Destrade
    • 1
    • 2
  • B. Mac Donald
    • 3
  • J. G. Murphy
    • 3
    • 1
    Email author
  • G. Saccomandi
    • 4
  1. 1.School of Mathematics, Statistics, and Applied MathematicsNational University of Ireland GalwayGalwayIreland
  2. 2.School of Mechanical and Materials EngineeringUniversity College DublinDublin 4Ireland
  3. 3.Centre for Medical Engineering ResearchDublin City UniversityDublin 9Ireland
  4. 4.Dipartimento di Ingegneria IndustrialeUniversità degli Studi di PerugiaPerugiaItaly

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